I know that the electrical generators have mechanical and electrical power losses. But does electrical or magnetic phenomena in a generator produce a counter torque? Or all the electrical power losses are related to current losses?
Of course having the generator produce electrical power puts a load (torque in opposition to rotation) on the shaft. Without such load, there would be no mechanical power transferred into the generator, and it would violate conservation of energy by producing electrical power.
Electric generators, which can be thought of as mechanical to electrical power converters, can be made reasonably efficient. Good ones are 90% efficient or more. In that case, 90% of the load on the shaft is due to the electrical power produced, and the other 10% is due to losses. These losses include mechanical friction, electrical losses in the wires, and magnetic losses in the core material and due to induced currents in unwanted places.